In this paper we have continued the programme in Bonchi
et al. [1]} of a coalgebraic theory of bisimulation of
weighted automata. We have generalized the formal framework to one based on
weighted automata on semirings rather than fields. As we have shown, almost all
the results of [1]} continue to hold except for the
partition refinement algorithm. The latter necessarily depends on the specific
type of semiring involved. Hence we can only propose an abstract procedure for
partition refinement that may not, in general, halt or lead to a solution even
if it halts, because of the the lack of enough operations.
